Given:
[tex]\tan \theta > 0[/tex] and [tex]\sec \theta <0[/tex].
To find:
The quadrant.
Solution:
We know that,
In I quadrant, all trigonometric ratios are positive.
In II quadrant, only [tex]\sin \theta, \text{cosec}\theta[/tex] are positive and others are negative.
In III quadrant, only [tex]\cot \theta, \tan\theta[/tex] are positive and others are negative.
In IV quadrant, only [tex]\cos \theta, \sec\theta[/tex] are positive and others are negative.
We have,
[tex]\tan \theta > 0[/tex] and [tex]\sec \theta <0[/tex].
Here, [tex]\tan \theta[/tex] is positive and [tex]\sec \theta[/tex] is negative. So, [tex]\theta [/tex] lies in the III quadrant.
Therefore, the correct option is C, i.e., III.
